MATH 1020: Mathematics For Non-science
Chapter 3: Information in a
networked age
Instructor: Dr. Ken Tsang
Room E409-R9
Email: kentsang@uic.edu.hk
1
Transmitting Information
– Binary codes
– Data compression
– Encoding with parity-check sums
– Cryptography
– Model the genetic code
2
Information , data & numbers

Today information are transmitted all over
the world through the internet
 Information is just collection of data
– Pictures – jpg, tif …
– Sound – mp3, mp4
– Video – wmv, mvb

Data consisted of numbers
3
Decimal Number System

As human normally counts with hands and there are
totally 10 fingers on both hands, this probably explains
the origin of the decimal number system.

10 digits:
– 0,1,2,3,4,5,6,7,8,9

Also called base-10 number system,
– Or Hindu-Arabic, or Arabic system

Counting in base-10
– 1,2,…,9,10,11,…,19,20,21,…,99,100,…

Decimal number in expanded notation
– 234 = 2 * 100
+ 3 * 10 + 4 * 1
4
Hindu–Arabic numeral system

The Brahmi (ancient Indian) numerals at the basis of the
system predate the Common Era.

The development of the positional decimal system occurred during
the Gupta period (笈多王朝, 320 to 540 CE).
Aryabhata, a Gupta period scholar, is believed to be the first to
come up with the concept of zero.


These Indian developments were taken up in Islamic
mathematics in the 8th century.

A young Italian in the 12th century, Fibonacci, traveled throughout the
Mediterranean world to study under the leading Arab mathematicians of the time,
recognizing that arithmetic with Hindu-Arabic numerals is simpler and more
efficient than with Roman numerals.
5
Fibonacci (1170-1250 CE)


Italian mathematician, Leonardo Fibonacci
(through the publication in 1202 of his Book of
Calculation, the Liber Abaci) introduced the
Arabic numerals, the use of zero, and the
positional decimal system to the Latin world.
Liber Abaci showed the practical importance
of the new numeral system, by applying it to
commercial bookkeeping.
The numeral system came to be called "Arabic" by the Europeans.
It was used in European mathematics from the 12th century, and
entered common use from the 15th century.
Fibonacci significantly influenced the evolution of capitalist
enterprise and public finance in Europe in the centuries that
followed.
6
Positional Numbering System
 The
value of a digit in a number
depends on:
– The digit itself
– The position of the digit within the number
 So
123 is different from 321
– 123: 1 hundred, 2 tens, and 3 units
– 321: 3 hundred, 2 tens, and 1 units
7
Roman numerals

Roman numerals are numeral system of
ancient Rome based on the letters of the
alphabet
 The first ten Roman numerals are I, II, III, IV,
V, VI, VII, VIII, IX, and X. (no zero)
 Tens: X; hundreds: C; thousands: M
 Non-positional: e.g.
– 321  CCCXXI
– 982  CMLXXXII
– 2010  MMX
8
Non-decimal Number Systems




The Maya civilization and other civilizations of preColumbian Mesoamerica used base-20 (vigesimal), as did
several North American tribes (two being in southern
California).
Evidence of base-20 counting systems is also found in the
languages of central and western Africa.
The Irish language also used base-20 in the past.
Danish numerals display a similar base-20 structure.
9
Base r Number System
 For
any value r
 Value is based on the sum of a
power series in powers of r
r
is called the base, or radix
10
Binary Number System

Binary number system has only two digits
– 0, 1
– Also called base-2 system

Counting in binary system
– 0, 1, 10, 11, 100, 101, 110, 111, 1000,….

Binary number in expanded notation
– (1011)2 = 1*23
– (1011)2 = 1*8
+ 0*22 + 1*21 + 1*20
+ 0*4 + 1*2 + 1*1 = (11)10
11
Why Binary?

Computer is a Binary machine
It knows only ones and zeroes
Easy to implement in electronic
circuits
Reliable
Cheap
12
Gottfried Leibniz (1646-1716)
Leibniz, German mathematician
and philosopher, invented at
least two things that are
essential for the modern world:
calculus, and the binary
system.
He invented the binary system
around 1679, and published in
1701. This became the basis of
virtually all modern computers.
13
Leibniz's Step Reckoner

Leibniz designed a machine to
carry out multiplication, the
'Stepped Reckoner'. It can multiple
number of up to 5 and 12 digits to
give a 16 digit operand. The
machine was later lost in an attic
until 1879.
14
Leibniz & I-Ching (易经)

As a Sinophile, Leibniz was aware of the IChing and noted with fascination how its
hexagrams correspond to the binary
numbers, and concluded that this mapping
was evidence of major Chinese
accomplishments in the sort of
philosophical mathematics he admired.
15
An ancient Chinese binary
number system in Yi-Jing (易经)

Two symbols to represent 2 digits
 Zero: represented by a broken line
 One: represented by an unbroken line
 “—” yan 阳爻，“--” yin 阴爻。
16
Hexadecimal
 Hexadecimal
number system has 16 digits
• 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
• Also called base-16 system
 Counting in Hexadecimal
– 0,1,…,F,10,11,…,1F,20,…FF,100,…
 Hexadecimal
number in expanded notation
– (FF)16 = 15*161
+ 15*16
0
= (255)10
17
Some Numbers to
Remember
18
Bit and Byte
 BIT
= Binary digIT, “0” or “1”
 State
of on or off ( high or low) of a
computer circuit
 Kilo
1K = 210 = 1024 ≈ 103
 Mega
 Giga
1M = 220 = 1,048,576 ≈ 106
1G = 230 = 1,073,741,824 ≈ 109
19
Bit and Byte

Byte is the basic unit of addressable memory
 1 Byte = 8 Bits
 The right-most bit is called the LSB
Least Significant Bit
 The Left-most bit is called the MSB
Most Significant Bit
20
Natural Numbers

Natural numbers
– Zero and any number obtained by repeatedly adding
one to it

Negative Numbers
– A value less than 0, with a – sign

Integers
– A natural number, a negative number, zero

Rational Numbers
– An integer or the quotient of two integers
 We
will only discuss the binary
representation of non-negative integers
21
Why Hexadecimal?
 Hexadecimal
is meaningful to humans,
and easy to work with for a computer
 Compact
– A BYTE is composed of 8 bits
– One byte can thus be expressed by 2
digits in hexadecimal
– 11101111  EF
– 11101111b  EFh
 Simple
to convert them to binary
22
Conversions
Between Number Systems

Binary to Decimal
23
Conversions
Between Number Systems

Hexadecimal to Decimal
24
Conversions
Between Number Systems
 Octal
to Decimal
– (32)8 = (?)10
 What’s
wrong?
– (187)8 = 1*64 + 8*8 + 7*1
25
Conversions
Between Number Systems

Decimal to Binary
32110 = ?2
quotient remainder
321 / 2 = 160
160 / 2 = 80
80 / 2 = 40
40 / 2 = 20
20 / 2 = 10
10 / 2 = 5
5/2 =2
2/2 =1
1/2 =0
1
0
0
0
0
0
1
0
1
from bottom to top, we have
Reading the remainders
32110 = 1010000012
26
One More Example
Convert 14710 to binary
So, 14710 = 100100112
27
Conversions
Between Number Systems
 Decimal
to Base r
– Same as Decimal to Binary
– Divide the number by r
– Record the quotient and remainder
– Divide the new quotient by r again
– …..
– Repeat until the newest quotient is 0
– Read the remainder from bottom to top
28
Analogue Data

Analogue: something that is analogous or
similar to something else (Webster)
 Analogue Data: The use of continuously
changing quantities to represent data.
 A mercury thermometer is an analogue
device. The mercury rises and falls in a
continuous flow in the tube in direct
proportion to the temperature.
 The mathematical idealization of this smooth
change as a continuous function leads to
“Analogue Data”, an infinite amount of data
29
From Analogue to Digital data

Data can be represented in one of two ways:
analogue or digital:
Analogue data: A continuous representation
(using mathematical function or smooth curve) ,
analogous to the actual information it represents
Digital data: A discrete representation, breaking
the information up into separate elements (data)
30
Digitized Information

Computers, cannot work with analogue
information
 So we digitize information by breaking it into
pieces and representing those pieces separately
 Why do we use binary?
– Modern computers are designed to use and manage
binary values because the devices that store and
manage the data are far less expensive and far more
reliable if they only have to represent one of two
possible values.
31
Binary Representation
One bit can be either 0 or 1 (“on” & “off”
electronic signals)
 Therefore, one bit can represent only two
things
 To represent more than two things, we need
multiple bits
 Two bits can represent four things because
there are four combinations of 0 and 1 that
can be made from two bits: 00, 01, 10, 11

32
Binary Representation
Represents
2 numbers
4
8
16
32
33
Binary Representation
n

In general, n bits can represent 2 things
n
because there are 2 combinations of 0 and 1
that can be made from n bits
 Note that every time we increase the number
of bits by 1, we double the number of things
we can represent

Questions:
– How many bits are needed to represent 128 things?
– How many bits are needed to represent 67 things?
34
Binary mathematics
Logical operations
AND
OR
XOR
35
ASCII
 ASCII
stands for American Standard
Code for Information Interchange
 The ASCII character set originally
used seven bits to represent each
character, allowing for 128 unique
characters
 Later ASCII evolved so that all eight
bits were used which allows for 256
characters
36
ASCII code
37
ASCII
 Note
that the first 32 characters in
the ASCII character chart do not
have a simple character
representation that you could print
to the screen (unprintable)
38
Unicode characters

Extended version of the ASCII character set is
not enough for international use
 The Unicode character set uses 16 bits per
character
– Therefore, the Unicode character set can represent 216, or over
65 thousand, characters

Unicode was designed to be a superset of ASCII
– The first 256 characters in the Unicode character set
correspond exactly to the extended ASCII character set

With the Unicode, all text (in most languages)
information can be represented.
39
4 Hex-numerals
to represent 1
Unicode
Unicode
40
Representing Audio
Information
 To
digitize the signal we periodically
measure the voltage of the signal and
record the appropriate numeric value
– this process is called sampling
 In
general, a sampling rate of around
40,000 times per second is enough to
create a reasonable sound reproduction
41
Representing Audio
Information
42
Representing Audio
Information
• A compact disk (CD) stores
audio information digitally
• On the surface of the CD are
microscopic pits that represent
Binary digits
•A low intensity laser is
pointed as the disc
•The laser light reflects
strongly if the surface is
smooth and reflects poorly if
the surface is pitted
43
Representing Audio
Information

Audio Formats
– WAV, AU, AIFF, VQF, and MP3

MP3 is dominant
– MP3 is short for MPEG (Moving Picture Experts Group)
audio layer 3 file
– MP3 employs both lossy and lossless compression
– First it analyzes the frequency spread and compares it to
mathematical models of human psychoacoustics (the
study of the interrelation between the ear and the brain),
then it discards information that can’t be heard by
humans
– Then the bit stream is compressed to achieve additional
compression
44
Image Basics
00000000000000000011110000000000000000
00000000000000001100001100000000000000
00000000000000010000000010000000000000
00000000000000100000000001000000000000
00000000000000100010001001000000000000
00000000000001000111011100100000000000
00000000000001000010001000100000000000
00000000000001000000000000100000000000
00000000000001000000000000100000000000
00000000000001001000000100100000000000
00000000000000100100001001000000000000
00000000000000100011110001000000000000
00000000000000010000000010000000000000
00000000000000001100001100000000000000
00000000000000000011110000000000000000
00011110010000000000000000000000000000
01100010010000000000000000000000000000
11000100100000000000000000000000000000
00000100100001110001011000101100100100
00111111110010010001101000110101100100
00001001000100100111001011100101001000
00010010000101101010010101001011011010
00010010000110110111111011111101101100
00000000000000000100000010000000011000
00000000000000001100000110000000110000
00000000000000001000000100000000100000
46
1-bit, black and white
8-bit grayscale
48
Representing Color
 Color
is often expressed in a computer as
an RGB (red-green-blue) value, which is
actually three numbers that indicate the
relative contribution of each of these
three primary colours
 For example, an RGB value of (255, 255,
0) maximizes the contribution of red and
green, and minimizes the contribution of
blue, which results in a bright yellow
49
RGB Model

RGB Color Model
– Red – Green – Blue
– Additive model combines varying amounts of
these 3 colors
50
Image Basics
(0, 255, 0) is green
(255, 255, 0)
is yellow
(255, 0, 0)
is red
(0, 255, 255)
is cyan
(0, 0, 255)
is blue
(0, 0, 0) is black
(255, 255, 255) is white
(255, 0, 255)
is magenta
53
Three Dimension Color
Space
(0,0,0)
(1,1,1)
54
Composing color image

Store the actual intensities of R, G, and B
individually in the framebuffer
 24 bits per pixel = 8 bits red, 8 bits green, 8
bits blue
DAC
55
Digitized Images and Graphics

Digitizing a picture is the act of representing it as
a collection of individual dots called pixels
 The number of pixels used to represent a picture
is called the resolution
 Several popular raster file formats including
bitmap (BMP), GIF, and JPEG
58
Image Basics

Bitmap
– Grid of pixels
59
BMP
60